Homework - Analytical Solution to Profile Model
In class we discussed the Dupuit problem with an infiltration term (e):
The equation for the flow rate as a function of x is:
\(Q = k\dfrac{\left(H_o^2-H_D^2\right)}{2D}-e\left(\dfrac{D}{2}-x\right)\)
Derive an equation for h in terms of x. Start with:
\(h^2 = -\dfrac{e}{k}x^2 + C1x + C2\)
and insert the left and right boundary conditions and solve for C1 and C2. Solve the resulting equation for h.
Put your solution in a Word document and show each of your steps. Or you can derive on paper and submit a scan or photo of your work.
Once you have found your equation for h, created a modified version of the solution to the Dupuit problem that we did in class that uses your new equation for h. You can do it in Python or Excel.
Excel starter file: dupuit_with_recharge.xlsx
Python starter file: 
Submission
Do not put your derivation in the Excel or Python file. It should be in a Word document or scan (photo or PDF).
If you completed the second part of the problem using Google Colab, use the "File|Download|Download .ipynb" command to download your notebook as an .ipynb file.
Create a zip archive with your Word document (or scan) and the solution file (either the *.xlsx Excel file or the .ipynb notebook file) and upload the zip archive to Learning Suite.
Grading Rubric
Total: 30 points
| Criteria | Points |
|---|---|
| Derivation: Applying left boundary condition | 4 |
| Derivation: Applying right boundary condition | 4 |
| Derivation: Solving for C1 | 3 |
| Derivation: Solving for C2 | 3 |
| Derivation: Final equation for h(x) | 4 |
| Implementation: Excel or Python solution correctly set up | 6 |
| Implementation: Results plotted and verified | 4 |
| Documentation and clarity of work | 2 |